On finite p-groups minimally of class greater than two

Let G be a finite nilpotent group of class three whose proper subgroups and proper quotients are nilpotent of class at most two. We show that G is either a 2-generated p-group or a 3-generated 3-group. In the first case the groups of maximal order with respect to a given exponent are all isomorphic except in the cases where p = 2 and exp(G) = 2(r), r >= 4. If G is 3-generated, then we show that there is a unique group of maximal order and exponent 3; but a similar result is not valid for exponent 9.